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Inequality with gcd

Source: Russian TST 2020, Day 6 P2

March 21, 2023
number theoryinequalitiesgreatest common divisor

Problem Statement

Given a natural number nn{} find the smallest λ\lambda such thatgcd⁡(x(x+1)⋯(x+n−1),y(y+1)⋯(y+n−1))⩽(x−y)λ,\gcd(x(x + 1)\cdots(x + n - 1), y(y + 1)\cdots(y + n - 1)) \leqslant (x-y)^\lambda, for any positive integers yy{} and x⩾y+nx \geqslant y + n.
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